Structured sparsity

To calculate numerically the quantum dynamics of the various cations in time-dependent domain, we shall use the multiconfiguration time-dependent Hartree method (MCTDH) [79-82, 113, 114]. This method for propagating multidimensional wave packets is one of the most powerful techniques currently available. For an overview of the capabilities and applications of the MCTDH method we refer to a recent book [114]. Additional insight into the vibronic dynamics can be achieved by performing time-independent calculations. To this end Lanczos algorithm [115,116] is a very suitable algorithm for our purposes because of the structural sparsity of the Hamiltonian secular matrix and the matrix-vector multiplication routine is very efficient to implement [6]. 

In addition, the non-zero elements follow a simple pattern, both by position in the matrix and by value. This behaviom has been termed structured sparsity and greatly facilitates the computations. It is a direct consequence of the concept of the low-order Taylor expansion introduced in Sec. 2 and becomes particularly pronounced for its simplest variant, the linear vibronic-coupling scheme, cf. Eq. (28). 

For medium and large networks, the occurrence matrix that is of the same structure (isomorphic) as the coefficient matrix of the governing equations is usually quite sparse. For example, Stoner (S5) showed a 155-vertex network with a density of 3.2% for the occurrence matrix (i.e., 775 nonzeros out of a total of 1552 entries) using formulation C. Still lower densities have been observed on larger networks. In these applications it is of paramount importance that the data structure and data manipulations take full advantage of the sparsity of the governing equations. Sparse computation techniques are also needed in order to capture the full benefit of cycle selection and row and column reordering. 

In the last decade, quantum-chemical investigations have become an integral part of modern chemical research. The appearance of chemistry as a purely experimental discipline has been changed by the development of electronic structure methods that are now widely used. This change became possible because contemporary quantum-chemical programs provide reliable data and important information about structures and reactivities of molecules and solids that complement results of experimental studies. Theoretical methods are now available for compounds of all elements of the periodic table, including heavy metals, as reliable procedures for the calculation of relativistic effects and efficient treatments of many-electron systems have been developed [1, 2] For transition metal (TM) compounds, accurate calculations of thermodynamic properties are of particularly great usefulness due to the sparsity of experimental data. 

Do not compute or use a matrix inverse unless absolutely needed for theoretical purposes do not ever destroy the sparsity of a matrix, nor increase the density of an intermediate matrix be mindful of the structure of each matrix and try to take advantage of it and preserve it during numerical computations. Rely on matrix factorizations, and view coupled matrix equations as block matrix equations before actually trying to solve them. 

There have been relatively few studies of oxide surfaces, primarily due to charging problems encountered when attempting to perform LEED measurements on these insulating surfaces. Because of the sparsity of structural information, it is difficult to establish any structural trends for oxide surfaces at present. However, a common feature of all determinations that have looked for it, is an approximately 0.1 A buckling of the top metal-O layer in which the O atom moves out of the surface. 

The other approach, taken by Hoffman-LaRoche, was to attempt to discover a new, chemically unique lead structural type. Considering the sparsity of knowledge and understanding of the chemical processes of the brain, much less their relationship to behavior and mental disease, such an undertaking might have been akin to a search for the Fountain of Youth. In practical terms, such discoveries had been made based on random screening with the expectation of an accidental hit or, less colloquially, a serendipitous discovery. Tests were developed to determine the ability of compounds to... 

By far the greatest number of literature citations are devoted to the pyrrolotriazines (1-6). Compounds of this type are the only representatives in this section which can form electronically neutral fully delocalized lOTt-electron systems. Most of the parent systems are known, but surprisingly, these have received scant attention in the literature, the greatest number of citations being devoted to the nonconjugated systems. There has been little systematic study of spectral properties, x-ray structure, and molecular dimensions determination, and the use of theoretical methods to predict reactivity at ring atoms. Thus, owing to the sparsity of information, little space is devoted here to these subjects unless some feature warrants special note. 

Very few published reports deal with structural changes in wood caused by aging. These reports are reviewed in the first part of this chapter. The sparsity of available information forced us to undertake further studies, which are reported in the second part of the chapter. The third part contains a brief review of the known structural changes in dry wood resulting from insect attack. 

Differential-algebraic systems are explored in greater depth in Chapter 4. Special algorithms to handle this family of problems are described and implemented in the BzzMath library. Classes to handle the sparsity and structure of such systems typical of chemical engineering are also described. 

The solution of the system (2.219) should be achieved by the method that best exploits Jacobian sparsity and structure. 

Using a DAE solver that accepts the Jacobian existence matrix, that is, an incidence matrix only. This provides the solver with the possibility to completely exploit the system s sparsity, but not its overall structure. Nevertheless, it is often not possible to provide the Jacobian incidence matrix, especially when the system is very large or when the incidence matrix changes as part of an iterative process. 

The first alternative is used when the matrix A2,i is sparse and the second alternative is used when the matrix A12 is sparse. The algorithm exploits the sparsity and structure of matrix Aii. It is consequently factorized just once. 

Procedure A BzzDae is used to solve the system. Neither structure nor sparsity information is provided. 

In turn, procedure B is four times slower than procedure C for all the step input signals studies, as the latter does not only exploit the system sparsity and structure, but also the location of the unstructured nonzero elements. By adopting the partially structured DAE solver, one needs half a second to simulate open-loop scenarios and about Is for the closed-loop ones. 

C. Sparsity, Structure, and Computer Representation of Linear Systems... 

The scaling is due to the structure of the product basis. It is not a consequence of the sparsity of the Hamiltonian matrix or the sparsity of the matrices... 

One of the main challenges in correctly using P2 and implementing 3 degree price discrimination is sparsity of data. Given the typically large number of products sold by the client, many of which have very sparse transaction histories, it is not possible to obtain robust estimates of the sensitivity parameter independently for each product. In order to deal with this, P2 makes use of market structure characteristics of each product to pool transactions for the purposes of estimation. The estimates obtained from this group of products are refined for each individual product, based on reliability measures from the estimation. 

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